Degenerate diffusion with a drift potential: A viscosity solutions approach
Inwon C. Kim Helen K. Lei
Discrete & Continuous Dynamical Systems - A 2010, 27(2): 767-786 doi: 10.3934/dcds.2010.27.767
We introduce a notion of viscosity solutions for a nonlinear degenerate diffusion equation with a drift potential. We show that our notion of solutions coincide with the weak solutions defined via integration by parts. As an application of the viscosity solutions theory, we show that the free boundary uniformly converges to the equilibrium as $t$ grows. In the case of a convex potential, an exponential rate of free boundary convergence is obtained.
keywords: well-posedness porous medium equation degenerate diffusion viscosity solutions convergence to equilibrium.
The supercooled Stefan problem in one dimension
Lincoln Chayes Inwon C. Kim
Communications on Pure & Applied Analysis 2012, 11(2): 845-859 doi: 10.3934/cpaa.2012.11.845
We study the 1D contracting Stefan problem with two moving boundaries that describes the freezing of a supercooled liquid. The problem is borderline ill--posed with a density in excess of unity indicative of the dividing line. We show that if the initial density, $\rho_0(x)$ does not exceed one and is not too close to one in the vicinity of the boundaries, then there is a unique solution for all times which is smooth for all positive times. Conversely if the initial density is too large, singularities may occur. Here the situation is more complex: the solution may suddenly freeze without any hope of continuation or may continue to evolve after a local instant freezing but, sometimes, with the loss of uniqueness.
keywords: Stefan problem supercooling phase transition
Inwon C. Kim
Discrete & Continuous Dynamical Systems - A 2011, 30(1): 375-377 doi: 10.3934/dcds.2011.30.375
The earlier paper [2] contains a lower bound of the solution in terms of its $L^1$ norm, which is incorrect. In this note we explain the mistake and present a correction to it under the restriction that the permeability constant $m$ satisfies $1< m <2$. As a consequence, the quantitative estimates on the convergence rate (Main Theorem (c) and Theorem 3.6 in [2] ) only hold for $1<\m<2$. For $\m\geq 2$ a partial convergence rate is obtained.
keywords: degenerate diffusion asymptotic convergence. Viscosity solutions

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